Generators for a module of vector-valued Siegel modular forms of degree 2

TL;DR

This paper characterizes and constructs generators for a specific module of vector-valued Siegel modular forms of degree 2, revealing its free structure and introducing generalized Rankin-Cohen operators.

Contribution

It provides a complete description of the module of vector-valued Siegel modular forms of degree 2 and odd weight, and introduces generalized Rankin-Cohen brackets for these forms.

Findings

The module is free over the ring of classical Siegel modular forms.

Explicit generators for the module are constructed.

A new Rankin-Cohen bracket for vector-valued forms is defined.

Abstract

In this paper we will describe all vector-valued Siegel modular forms of degree 2 and weight ${\rm Sym}^6({\rm St}) \otimes \det^{k}({\rm St})$ with $k$ odd. These vector-valued forms constitute a module over the ring of classical Siegel modular forms of degree 2 and even weight and this module turns out to be free. In order to find generators, we generalize certain Rankin-Cohen differential operators on triples of classical Siegel modular forms that were first considered by Ibukiyama and we find a Rankin-Cohen bracket on vector-valued Siegel modular forms.